contour integration?

[galactus]

Happy Thanksgiving to all.

Anyway, I was wondering if someone could recommend a good complex analysis text.

For instance, here's a problem I seen on an old GRE exam. I haven't learned much about this sort of integration.

In the complex plane, let C be the circle |z|=2 with positive orientation, evaluate:

\(\displaystyle \L\\\int_{C}\frac{dz}{(z-1)(z+3)^{2}}\)

I know the answer is \(\displaystyle \frac{\pi}{8}i\). How is that arrived at?.

If it's too much to go into, don't bother. I just wanted an idea to get me started.

 

[pka]

Using the Cauchy Integral Formula we get:
\(\displaystyle \L\int\limits_C {\frac{{dz}}{{\left( {z - 1} \right)\left( {z + 3} \right)^2 }}} = \int\limits_C {\frac{{\left( {z + 3} \right)^{ - 2} dz}}{{\left( {z - 1} \right)}} = 2\pi i\left( {1 + 3} \right)^{ - 2} }.\)
You see the singularity –3 is not on or interior to the contour.

I recommend Basic Complex Variables by John H. Mathews. It is written as an applied mathematics text. But at the same time it is complete. It is fairly up to date on notations usage.

 

[galactus]

Thank you pka. I understand that. I will look up that book.

Don't eat too much . I will.

 

[marcmtlca]

I used "Complex Variables with applications" by A. David Wunsch. But it is awful.

 

[pka]

I used "Complex Variables with applications" by A. David Wunsch. But it is awful.

I agree! I don't know why or how it got published.